Internal Heat Generation Parameter Research Articles

A neural network approach to modeling magnetohydrodynamic stagnation point Ree-Eyring flow over a convectively heated stretched surface

ABSTRACT This study investigates the application of artificial intelligence (AI) in fluid dynamics, mainly using a neural network trained by the Levenberg – Marquardt method (NN-BLMM), to model magnetohydrodynamic (MHD) stagnation point Ree – Eyring flow. We focus on this flow over a convectively heated stretched surface by integrating the Cattaneo – Christov heat model. The initial complex nonlinear partial differential equations (PDEs) are transformed into ordinary differential equations (ODEs) using suitable similarity variables. A dataset was generated using the Lobatto IIIA numerical solver to analyze the effects of various fluid flow and thermal parameters. The NN-BLMM model was then rigorously evaluated through training, testing, and validation phases and compared with reference data. This ensures the model’s precision and effectiveness. We observe that an increase in the Powell – Eyring fluid parameter notably reduces the fluid’s shear resistance, implying a decrease in viscosity. Concurrently, the heat transfer rate within the fluid medium increases with an increase in the internal heat generation parameter. These findings highlight the robustness of NN-BLMM in simulating complex fluid dynamics, emphasizing AI’s potential to provide a deeper understanding of non-Newtonian fluid behaviors. This research has important implications for industrial applications in which precise control over fluid properties and heat transfer is crucial.

Hydrothermal characteristics of ferrofluid in a wavy chamber with magnetic field-dependent viscosity: Effects of moving walls

The present investigation aims to scrutinize the role of different moving walls on the hydrothermal characteristics and thermal performance of a Fe3O4-H2O ferrofluid within a wavy chamber under mixed convection. Notably, the study also takes into consideration the effects of a magnetic field and internal heat generation or absorption phenomena. To achieve this, a numerical simulation employing a compact finite difference method is conducted, focusing on key factors such as the Richardson number (0.1≤Ri≤100), Hartmann number (0≤Ha≤30), magnetic orientation (00≤γ≤900), Grashof number (Gr=104), solid concentrations (0≤ϕnp≤0.04), magnetic number (0≤δ0≤1), Prandtl number (Pr=6.8377), and internal heat generation parameter (−2≤Q≤2). All of this is undertaken while carefully managing the ferromagnetic particle volume fraction. The primary aim is to assess how different types of moving walls affect the transport behavior of the ferrofluid and to determine the impacts of the magnetic field on the hydrothermal characteristics of the ferrofluid. The outcomes of this investigation reveal that the presence of moving walls leads to a non-uniform temperature distribution within the ferrofluid, significantly affecting the overall transport behavior. The addition of ferrofluid particles to the base fluid results in noteworthy enhancements in the average Nusselt numbers, signifying improved convective heat transfer efficiency. Specifically, Case-I experienced an increase of up to 3.21%, Case-II showed an improvement of 0.93%, Case-III exhibited a rise of 1.70%, and Case-IV demonstrated the highest enhancement of 20.71%. Furthermore, the inclusion of magnetic field-dependent viscosity and the internal heat generation coefficient significantly influences the behavior of the ferrofluid. The findings from this study hold paramount importance for the design and optimization of heat transfer systems that employ ferrofluids and have the potential to contribute to the development of novel technologies with a wide range of industrial applications.

Entropy generation on magnetohydrodynamic conjugate free convection with Joule heating of heat-generating liquid and solid element inside a chamber

This study presents the entropy production of MHD conjugate free convective flow within a square chamber infused with water encasing a heat-generating and conducting square solid element. The chamber’s vertical side is exposed to an external magnetism effect. The flowing fluid circulating inside the hollow space of the chamber produces a volumetric heat generation rate. The finite element approach is used to solve the model equations for mass, momentum, and heat energy conservations, which are affected by a number of variables, including the Rayleigh number (Ra), Hartmann number (Ha), Joule heating parameter, and internal heat generation parameter (Δ), and the area ratio of the solid element (Ab). By systematically varying those parameters (e.g., 103 ≤ Ra ≤ 106, 0 ≤ Ha ≤ 20, 0 ≤ Δ ≤ 10, 0.1 ≤ Ab ≤ 0.3), variations of hydro-thermal properties are observed and analyzed through streamline, isotherm, Nusselt number, entropy generation, thermal performance criterion, and mean fluid temperature of the system. It can be declared that convective heat transport increases as the area ratio of the solid element and the internal heat generation decrease. However, the minimum thermodynamic irreversibility is obtained for the highest area ratio and lowest heat generation parameter. The study shows a 48.37% increase in convective heat transfer as Ab decreases from 0.3 to 0.1. Similarly, reducing Ha and Δ to zero leads to an 8.76% and 526% reduction in TPC, respectively.

Numerical Study of Natural Convection Flow in Rectangular Cavity with Viscous Dissipation and Internal Heat Generation for Different Aspect Ratios

Numerical simulations have been performed to investigate the influence of constant volumetric heat generation and viscous dissipation on the unsteady natural convection flow of an incompressible Newtonian fluid contained in a rectangular cavity. The left vertical wall of the cavity is cooled, while the right vertical wall is heated, and the bottom and top walls are adiabatic. A numerical technique based on the implicit finite difference method (IFDM), along with an upwind finite difference scheme and an iterative successive over relaxation (SOR) technique, is employed to solve the governing equations numerically. The effect of physical parameters, namely the modified Rayleigh number (103≤Ra≤107), aspect ratio (1≤A≤4), Prandtl number (Pr=0.7, 1.0, 6.2, 15), volumetric internal heat generation parameter (Qλ=0, 1), and Eckert number (0≤Ec≤10−6), on the streamlines and isotherms are discussed graphically. Variations of maximum stream function, as well as average and local Nusselt number, are also discussed. The results show that the increase in Eckert number from 0 to 10−4 causes the average heat transfer to decrease, while Pr=0.71, Ra=104, and Qλ=0. Additionally, the average heat transfer decreases as the cavity width increases from 1 to 4, while Pr=0.71, Ra = 5×104, Ec=10−6 and Qλ=1. The results of the numerical model used here are in excellent accord with earlier findings.

Convectively cooled solidification in phase change materials in different configurations subject to internal heat generation: Quasi-steady analysis

Phase change phenomena involving internal heat generation are of central interest in various applications, including freeze drying, preservation of biological tissues, microwave/Joule heating processes, materials processing using lasers, and nuclear power generation systems. In this work, mathematical models are developed to study the solidification processes driven by convective cooling effects on the exterior surface of phase change materials (PCM) in different geometric configurations, where both the solid and the liquid phases are subject to volumetric heating. By invoking a quasi-steady assumption and performing a simplified unified analysis, new semi-analytical approximate solutions for this unique Stefan problem in planar, cylindrical, spherical, and semi-infinite geometries of the PCM are obtained and studied. The interface evolution depends on the characteristic dimensionless parameters, such as the Biot number, Bi, internal heat generation parameter, Q, Stefan number, St, and the thermal conductivity ratio, κ. The effects of all the dimensionless parameters, viz., Bi, Q, St, and κ are systematically investigated for each geometrical configuration of the PCM. In particular, the results show the significant role of the Biot number on the transient evolution as well as the steady-state thickness of the solidifying front in the presence of an inhibiting internal heat generation. In addition, a simple and unified condition relating the dimensionless internal heating rate (Q) and the external convective cooling rate (Bi) for the onset of remelting for the special case of κ = 1 applicable for all the above geometries of the PCM is deduced.

Unsteady magneto-convective heat-mass transport passing in a vertical permeable sheet with internal heat generation effect

The unsteady magneto-convective heat-mass transport passing in a vertical permeable sheet has been discussed with the internal heat generation/absorption effect. The governing partial differential equations (PDEs) have been transferred into ordinary differential equations (ODEs) by applying the local similarity transformation. The dimensionless ODEs are solved numerically by exerting the Quasilinearization Method with the help of ‘MATLAB ODE45’software. The influence of emerging dimensionless parameters/numbers for example the internal generation/absorption (Q), the suction parameter (v0), Prandtl number (Pr), Soret number (So), Dufour number (Df), Schmidt number (Sc), magnetic force parameter (M), local modified Grashof number (Gm) and local Grashof number (Gr) on the flow, temperature, and concentration profiles are presented and discussed. The fluid velocity and temperature enhance rising values of the internal heat generation and reduces the concentration but the reverse tends to be found for absorption. The velocity field reduces for increasing values of the suction parameter, Magnetic parameter, Prandtl number, and Schmidt number. The local skin friction increases by about 18% and reduces by 6% due to an increase in the internal heat generation parameter (1.0-3.0) and heat absorption parameter (3.0-10.0), respectively. Uplifting values of the heat absorption parameter (3.0-10.0) and internal heat parameter (1.0-3.0) enhance the heat transfer rate by 56% and decrease by about 42%, respectively. The mass transfer rate enhances by about 59% and decreases by 85% due to an increase in the internal heat generation parameter (1.0-3.0) and heat absorption parameter (3.0-10.0), respectively. Finally, we compared our numerical results with previously published paper and observed them be a excellent agreement.

Numerical analysis of hydromagnetic mixed convective flow in an internally heated vertical porous layer using thermal nonequilibrium model

AbstractThe inertial and viscous effects on mixed hydromagnetic convection in a vertical porous channel using a local thermal nonequilibrium model with uniformly distributed internal sources are analyzed. The flow in the porous medium is described by the Brinkman–Forchheimer extension of Darcy's momentum equation. The vertical rigid boundaries are maintained at constant but different temperatures and a uniform magnetic field is applied across the porous layer. The governing equations are solved numerically by the finite element method and analytically by the perturbation method. The influence of governing parameters on the flow variables and heat transfer is discussed. An extensive study designates that an increase in the Darcy number enhances the velocity distribution due to increased permeability of the medium while the opposite trend is observed with increasing Hartmann number due to the increased retarding nature of Lorentz force. The presence of an external heat source has a notable impact on the temperature gradient of the fluid, which results in an increase in temperature distribution and thermal state of the fluid. A quantitative increase in heat transfer coefficient results in an increase of fluid temperature toward the electrically nonconducting wall, and as the fluid moves toward the electrically conducting wall the fluid temperature decreases. The results indicate that the Nusselt number declines with an increase in the solid internal heat generation parameter.

Numerical study of magnetohydrodynamic natural convection in a non-Darcian porous enclosure filled with electrically conducting helium gas

A theoretical and computational study of MHD natural convection in an isotropic non-Darcian porous medium saturated with electrically conducting helium gas in an enclosure in the presence of heat generation is presented. A Brinkman extended Darcy-Forchheimer model is employed and the working fluid is assumed to be incompressible. The model is non-dimensionalised and converted into pressure-velocity form. The Harlow-Welch marker and cell (MAC) finite difference technique is employed to solve the nonlinear boundary value problem via pressure-vorticity coupling. A parametric investigation of the influence of Grashof number ( Gr), Hartmann magnetic number ( Ha), Darcy number ( Da), and the internal heat generation parameter ( Γ) on streamline and isotherm distributions with Prandtl number ( Pr) is 0.71 (Helium) is conducted. The variation in local Nusselt number along the left and right walls of the computational 2 D enclosure is also studied. Validation house-computational numerical MATLAB code is tests are included. Local Nusselt number is elevated at both left and right walls with greater Darcy number (higher medium permeability) and Grashof number. However, with greater internal heat generation, local Nusselt number magnitudes are enhanced at the left (cold) wall only but suppressed at the right (hot) wall. Increasing magnetic field reduces local Nusselt number at both left and right walls. With increasing magnetic field, the single vortex is strongly distorted and skewed towards the top left and lower right corners of the enclosure. Temperature contours at the left and right wall are however less intense with greater magnetic field effect. The simulations are of relevance to hybrid electromagnetic gaseous fuel cells, magnetic field control of filtration processes and porous media materials processing systems.

Flow Reversal of Fully Developed Combined Convection in a Vertical Channel with Boundary Condition of a Third Kind

The heat transfer process of combined convection and its flow pattern in a vertical channel is important, especially in environmental, industrial and engineering applications. There has been some concern regarding the heat transfer process in which it is difficult, expensive and time consuming. There are three main objectives to satisfy the purpose of this study which are to determine the impact on flow and heat transfer of the Robin temperature boundary condition, to verify whether the viscous dissipation plays a classical role in fluid flow and heat transfer and lastly to compare the degree of heat transfer facilitated by the boundary conditions of Dirichlet, Neumann and Robin. In this study, the effects of different dimensional parameters were tested and the flow reversal phenomenon was discussed. The Boundary Value Problem (BVP) was solved numerically using Maple using a built-in routine, dsolve. Validation study to the previously published problem was carried out to verify the accuracy of the present computation. The transformation of partial differential equation to ordinary differential equation involved similarity technique. The numerical results of the flow and temperature variables were presented graphically. The flow reversal occurred when the value of internal heat generation (G) and combined convection parameter (λ) were large. There was no occurrence of flow reversal when the values of local heating exponent (p) and the Biot Numbers (Bi) increase. However, the value of mass transfer coefficient (N) was different, gave no significant impact.

Effect of electromagnetic field on the thermal performance of longitudinal trapezoidal porous fin using DTM–Pade approximant

AbstractThe heat transfer and thermal distribution through porous fins have gotten a lot of attention in recent years due to their extensive applications in the manufacturing and engineering field. In porous fins, the impact of magnetic field aids in improved heat transfer enhancement. Also, the combination of an electric effect and a magnetic field considerably enhances heat transfer. In this direction, the thermal distribution through a convective–radiative longitudinal trapezoidal porous fin with the impact of an internal heat source and an electromagnetic field is discussed in the present analysis. The governing heat equation is nondimensionalized with nondimensional terms, and the transformed nonlinear ordinary differential equation is solved analytically using the DTM–Pade approximant algorithm. Furthermore, the graphical discussion is presented to explore the impact of various nondimensional parameters, such as convection‐conduction parameter, fin taper ratio, thermomagnetic field, radiation–conduction parameter, internal heat generation parameter, and thermoelectrical field on the temperature gradient of the fin. The investigation's key findings disclose that as the magnitude of the convection–conduction parameter, fin taper ratio, and radiation–conduction parameter increase, the thermal distribution through the fin reduces. The thermal distribution inside the fin increases for the heat‐generating parameter, thermoelectric, and thermomagnetic fields.

Thermal distribution through a moving longitudinal trapezoidal fin with variable temperature-dependent thermal properties using DTM-Pade approximant

The present inspection elaborates the thermal distribution through a convective-radiative longitudinal trapezoidal moving fin with an internal heat source by considering the linear and nonlinear temperature-dependent thermal conductivities and heat transfer coefficient. The non-dimensionalization of the energy equation is carried out using dimensionless terms, and the resulting nonlinear ordinary differential equation (ODE) is solved analytically using the differential transform method (DTM) with Pade approximant approach. Furthermore, the consequence of several non-dimensional parameters, namely thermal conductivity parameter, convection-conduction parameter, Peclet number, radiation-conduction parameter, fin taper ratio, temperature ratio parameter, and internal heat generation parameter on temperature gradient and fin efficiency is explained with the help of graphs. The significant primary results of the investigation illustrate that the thermal distribution through the fin gradually decreases as the magnitude of the convection-conduction parameter, temperature ratio parameter, and radiation-conduction parameter rise. In contrast, it upsurges for heat generation and thermal conductivity parameters.

Dynamics of hybrid nanofluid through a semi spherical porous fin with internal heat generation

Semi-spherical fins are widely considered to be one of the best thermal exchangers available. It is widely employed, including aero planes, chemicals and electronic kits and porous medium is an extensive application that includes drying efficiency enhancement, filtering insulations, hydraulic oils, reactor temperature control and solar collectors. With these applications in perspective, the current study uses Darcy’s model to examine the performance of hybrid nanofluids flowing over a semi-spherical porous fin. Additionally, temperature dependent internal heat generating condition, natural convection and radiation effects are also taken into account. The nonlinear form of an ordinary differential equation (ODE) along with boundary conditions is resolved using RKF 45 method. The numerical results are obtained for various physical constraints and the impact of these constraints on fin surface temperature is discussed via graphs. The results show that increasing an internal heat generation parameter raises temperature, whereas the opposite is true for radiation and convective parameters. Furthermore, the temperature of the hybrid nanofluid is consistently higher than the temperature of the nanofluid.

Significance of the inconstant viscosity and internal heat generation on the occurrence of Darcy-Brinkman convective motion in a couple-stress fluid saturated porous medium: An analytical solution

In this analysis, the collective effect of temperature-reliant viscosity and internal heat generation on the appearance of convective motion in couple-stress fluids saturated a thin porous layer is investigated employing linear stability principle. The significant influences of viscosity deviation parameter Q, internal heat generation parameter HS, couple-stress parameter Cs and Darcy number Da on the stability of the arrangement are obtained analytically and presented graphically in term of the critical thermal Darcy-Rayleigh number Rt,c. Below the critical values, no convective motion arises in the considered system. It is recognized that the viscosity deviation parameter Q and internal heat generation parameter HS have destabilizing effect, while the couple-stress parameter Cs and the Darcy number Da show stabilizing effect on the stability of the fluidic system. By increasing the couple-stress parameter Cs from 0.2 to 0.4 and the Darcy number Da from 0.25 to 0.50, the critical thermal Darcy-Rayleigh number Rt,c increased approximately 21% and 51%, respectively.

Effect of thermal non-equilibrium and internal heat source on Brinkman–Bénard convection

In the paper, we make a linear stability analysis in a Newtonian, liquid-saturated high porous medium in the presence of a heat source under the assumption of local thermal non-equilibrium(LTNE). The heat source is represented by an internal Rayleigh number RI and is considered in two ways (i) internal heat source in a liquid phase (RIl) and (ii) internal heat source in a solid phase (RIs). The internal heat generation parameter’s effect is to destabilize the onset of stationary convection due to an increase in the system’s energy. The results of local thermal equilibrium are obtained as a limiting case of the problem. The effect of various parameters on the onset of convection is analyzed and depicted graphically.

Analytical Analysis of Effects of Buoyancy, Internal Heat Generation, Magnetic Field, and Thermal Radiation on a Boundary Layer over a Vertical Plate with a Convective Surface Boundary Condition

In this paper, the effects of magnetic field, thermal radiation, buoyancy force, and internal heat generation on the laminar boundary layer flow about a vertical plate in the presence of a convective surface boundary condition have been investigated. In the analysis, it is assumed that the left surface of the plate is in contact with a hot fluid, whereas a stream of cold fluid flows steadily over the right surface, and the heat source decays exponentially outwards from the surface of the plate. The governing nonlinear partial differential equations have been transformed into a set of coupled nonlinear ordinary differential equations with the help of similarity transformation which were solved analytically by applying the optimal homotopy asymptotic method. The variations of fluid velocity and surface temperature for different values of the Grashof number, magnetic parameter, Prandtl number, internal heat generation parameter, Biot number, and radiation absorption parameter are tabulated, graphed, and interpreted in physical terms. A comparison with previously published results on similar special cases of the problem shows an excellent agreement.

Numerical simulation of hydromagnetic Marangoni convection flow in a Darcian porous semiconductor melt enclosure with buoyancy and heat generation effects

We present a mathematical and numerical study of the transient Marangoni thermo-convection flow of an electrically conducting Newtonian fluid in an isotropic Darcy porous rectangular semiconductor melt enclosure with buoyancy and internal heat generation effects, in an (x, y) coordinate system. The governing equations comprising the mass conservation, x-direction momentum, y-direction momentum and energy equation are formulated subject to a quartet of boundary conditions at the four walls of the enclosure. The upper enclosure wall is assumed to be “free” with an appropriate surface tension dynamic boundary condition. A series of transformations are implemented to render the mathematical model dimensionless and into a vorticity form. The governing thermophysical parameters are shown to be the Marangoni number for surface tension (thermocapillary) effects (Ma), Prandtl number (Pr), Grashof number for buoyancy effects (Gr), enclosure aspect ratio (A), Hartmann hydromagnetic number (Ha), Darcy number for bulk porous resistance (Da), and the internal heat generation parameter (Γ) the latter being a function of the internal (RaI) and global Rayleigh numbers (Ra). An efficient finite difference numerical method is employed to solve the boundary value problem. Validations with earlier purely fluid solutions (Da → ∞) are included. A mesh-independence test is included with further validation with other published studies. Isotherms and isovels (streamlines) are computed as are Nusselt numbers at selected boundaries. Solutions for the case of Pr = 0.054 (semiconductor melt) are also compared with earlier studies showing excellent correlation. The model finds applications in the bulk crystal growth of semiconductors, electromagnetic materials processing control and also hybrid fuel cells.

Natural convection and entropy production in hybrid nanofluid filled-annular elliptical cavity with internal heat generation or absorption

This study is an attempt to understand the characteristics of heat transfer by natural convection, flow, and entropy generation of Cu-Al2O3/H2O based hybrid nanoliquid filled-annulus delimited by two elliptic cylinders considering internal heat generation or absorption (IHG/A) phenomenon. The buoyancy-driven flow is induced by a thermal gradient between isothermal and differentially heated inner and outer cylinders. A numerical solution of the governing equations in the dimensionless and non-primitive form is performed using the technique of finite volume discretization. Impacts of diverse parameters of the study such as copper-alumina nanoparticles volumic concentration, Rayleigh number, and dimensionless internal heat generation or absorption parameter on the thermohydrodynamic characteristic and entropy generation are examined. An analysis of the results showed that the combined effects of internal heat generation/absorption and hybrid nanoliquid significantly alter the hydrothermal characteristics, heat transfer rate, and entropy generation within the annulus.

A Comparison of the Effects of Solar Radiation on Sandy-Loam and Clay-Loam Soils with Convective Boundary Condition

A comparison of the effects of solar radiation on sandy-loam and clay-loam soils with convective boundary condition was made. The dimensional governing equations (the continuity, momentum and energy equations) were converted to non-dimensional form using some set of dimensionless quantities, which resulted to a pair of partial differential equations of second order. With the use of perturbation method, these equations were reduced to ordinary differential equation and then solved analytically. Using the numerical values of the thermal conductivities of sandy-loam and clay-loam soils, a good comparison was made between the effects of the solar radiation parameter and internal heat generation parameter on the two soil samples. These results were worked out with the use of Matlab R2009b and visibly displayed on graphs for clearer illustration. For a rise in the intensity of solar radiation and mounting internal heat, the temperature of the soils increased generally. However, the proportion of moisture content influenced the level and the rate at which the temperature is boosted in both soil samples.

Thermoelastic study of a functionally graded annular fin with variable thermal parameters using semiexact solution

In this paper, the thermoelastic behavior of a functionally graded material (FGM) annular fin is investigated. The material properties of the annular fin are assumed to vary radially. The heat transfer coefficient and internal heat generation are considered to be functions of temperature. A closed form solution of nonlinear heat transfer equation for the FGM fin is obtained using the homotopy perturbation method (HPM) which leads to nonuniform temperature distributions within the fin. The temperature field is then coupled with the classical theory of elasticity and the associated thermal stresses are derived analytically. For the correctness of the present closed form solution for the stress field, the results are compared with the ANSYS-based finite element method (FEM) solution. The present HPM-based closed form solution of the stress field exhibits a good agreement with the FEM results. The effect of various thermal parameters such as the thermogeometric parameter, conduction-radiation parameter, internal heat generation parameter, coefficient of variation of thermal conductivity, and the coefficient of thermal expansion on the thermal stresses are discussed. The results are presented in both nondimensional and dimensional form. The dimensional stress analysis discloses the suitability of FGM as the fin material in practical applications.

Mixed convective magnetonanofluid flow over a backward facing step and entropy generation using extended Darcy–Brinkman–Forchheimer model

This article focuses on a numerical study for the effect of porosity and internal heat generation/absorption parameters on the mixed convective flow of nanofluid over a backward facing step along with the entropy generation. The channel downstream bottom wall is isothermally heated while the remaining walls of the channel are thermally insulated. The dimensionless governing equations are discretized utilizing the Galerkin-based finite element method. The discrete systems of nonlinear algebraic equations are computed using the Newton method and the corresponding linearized systems are treated using the monolithic geometric multigrid solver. Effect of different physical parameters in the specified ranges such as \((10 \le Re \le 200)\), \((0.01 \le Ri \le 20)\), \((0\le Ha \le 100)\), \((-10 \le q \le 10)\), \((10^{-6} \le Da \le 10^{-3})\), \((0.2 \le \epsilon \le 0.8)\) and \((0^\circ \le \gamma \le 180^\circ )\) on the fluid flow is analyzed with the help of the streamlines, isotherm patterns, and various graphs. It is noticed that the average heat transfer increases with the porosity parameter and reduces with the internal heat generation parameter. Furthermore, the entropy generation produced by the heat transfer and fluid friction is found to amplify with the porosity parameter, whereas a decline is recorded in it due to the magnetic field.